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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpaddat | Structured version Visualization version GIF version |
Description: Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.) |
Ref | Expression |
---|---|
paddfval.l | ⊢ ≤ = (le‘𝐾) |
paddfval.j | ⊢ ∨ = (join‘𝐾) |
paddfval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
paddfval.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
elpaddat | ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑆 ≤ (𝑝 ∨ 𝑄)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1184 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → 𝐾 ∈ Lat) | |
2 | simpl2 1185 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → 𝑋 ⊆ 𝐴) | |
3 | simpl3 1186 | . . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → 𝑄 ∈ 𝐴) | |
4 | 3 | snssd 4655 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → {𝑄} ⊆ 𝐴) |
5 | simpr 485 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅) | |
6 | snnzg 4623 | . . . 4 ⊢ (𝑄 ∈ 𝐴 → {𝑄} ≠ ∅) | |
7 | 3, 6 | syl 17 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → {𝑄} ≠ ∅) |
8 | paddfval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
9 | paddfval.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
10 | paddfval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
11 | paddfval.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
12 | 8, 9, 10, 11 | elpaddn0 36488 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ {𝑄} ⊆ 𝐴) ∧ (𝑋 ≠ ∅ ∧ {𝑄} ≠ ∅)) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 ∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟)))) |
13 | 1, 2, 4, 5, 7, 12 | syl32anc 1371 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 ∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟)))) |
14 | oveq2 7031 | . . . . . . 7 ⊢ (𝑟 = 𝑄 → (𝑝 ∨ 𝑟) = (𝑝 ∨ 𝑄)) | |
15 | 14 | breq2d 4980 | . . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑆 ≤ (𝑝 ∨ 𝑟) ↔ 𝑆 ≤ (𝑝 ∨ 𝑄))) |
16 | 15 | rexsng 4527 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → (∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟) ↔ 𝑆 ≤ (𝑝 ∨ 𝑄))) |
17 | 3, 16 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟) ↔ 𝑆 ≤ (𝑝 ∨ 𝑄))) |
18 | 17 | rexbidv 3262 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (∃𝑝 ∈ 𝑋 ∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟) ↔ ∃𝑝 ∈ 𝑋 𝑆 ≤ (𝑝 ∨ 𝑄))) |
19 | 18 | anbi2d 628 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → ((𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 ∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟)) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑆 ≤ (𝑝 ∨ 𝑄)))) |
20 | 13, 19 | bitrd 280 | 1 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑆 ≤ (𝑝 ∨ 𝑄)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 ∃wrex 3108 ⊆ wss 3865 ∅c0 4217 {csn 4478 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 lecple 16405 joincjn 17387 Latclat 17488 Atomscatm 35951 +𝑃cpadd 36483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-1st 7552 df-2nd 7553 df-lub 17417 df-join 17419 df-lat 17489 df-ats 35955 df-padd 36484 |
This theorem is referenced by: elpaddatiN 36493 elpadd2at 36494 pclfinclN 36638 |
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